LINEAR PROGRAMMING Example 1 Maximise I = x + 0.8y subject to x + y 1000 2x + y 1500 3x + 2y.
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LINEAR PROGRAMMING
y
960 800
Example 1
Maximise I = x + 0.8y subject to x + y 1000 2x + y 1500 3x + 2y 2400 640 480 320 160 0 0 160 320 480 640 800 960
x
Initial solution:
I = 0
at (0, 0)
LINEAR PROGRAMMING
Maximise subject to I = x + 0.8y x + y 2x + y 3x + 2y 1000 1500 2400
Example 1
Maximise where subject to
I
I - x - 0.8y x + y + s 1 2x + y 3x + 2y + s 2 + s 3 = 0 = 1000 = 1500 = 2400
SIMPLEX TABLEAU
I
1
0 0 0
x
-1 1 2 3
y
-0.8
1 1 2
s
1 0
1
0 0
Initial solution
s
2 0 0
1
0
s
3 0 0 0
1
RHS
0 1000 1500 2400
I = 0, x = 0, y = 0, s 1 = 1000, s 2 = 1500, s 3 = 2400
PIVOT 1
I
1 0 0 0
Choosing the pivot column
x
-1 1 2 3
y
-0.8
1 1 2
s
1 0 1 0 0
s
2 0 0 1 0
s
3 0 0 0 1 RHS 0 1000 1500 2400 Most negative number in objective row
I
1 0 0 0
PIVOT 1
x
-1 1 2 3
Choosing the pivot element
y
-0.8
1 1 2
s
1 0 1 0 0
s
2 0 0 1 0
s
3 0 0 0 1 RHS 0 1000 1000/1 1500 1500/2 2400 2400/3 Ratio test: Min. of 3 ratios gives 2 as pivot element
PIVOT 1
I
1 0
0
0
Making the pivot
x
-1 1
1
3
y
-0.8
1
0.5
2
s
1 0 1
0
0
s
2 0 0
0.5
0
s
3 0 0
0
1 RHS 0 1000
750
2400 Divide through the pivot row by the pivot element
PIVOT 1
I
1
0 0 0
Making the pivot
x
0
1 1 3
y
-0.3
1 0.5
2
s
1
0
1 0 0
s
2
0.5
0 0.5
0
s
3
0
0 0 1 RHS
750
1000 750 2400 Objective row + pivot row
PIVOT 1
I
1
0
0 0
x
0
0
1 3
Making the pivot
y
-0.3
0.5
0.5
2
s
1 0
1
0 0
s
2 0.5
-0.5
0.5
0
s
3 0
0
0 1 RHS 750
250
750 2400 First constraint row - pivot row
PIVOT 1
I
1 0 0
0
x
0 0 1
0 Making the pivot
y
-0.3
0.5
0.5
0.5
s
1 0 1 0
0
s
2 0.5
-0.5
0.5
-1.5
s
3 0 0 0
1
RHS 750 250 750
150
Third constraint row – 3 x pivot row
PIVOT 1
I
1
0 0 0
x
0 0
1
0
New solution
y
-0.3
0.5
0.5
0.5
s
1 0
1
0 0
s
2 0.5
-0.5
0.5
-1.5
s
3 0 0 0
1
RHS
750 250 750 150
I = 750, x = 750, y = 0, s 1 = 250, s 2 = 0, s 3 = 150
LINEAR PROGRAMMING
y
960 800
Example
Maximise I = x + 0.8y subject to x + y 1000 2x + y 1500 3x + 2y 2400 640 480 320 160 0 0 160 320 480 640 800 960
x
Solution after pivot 1:
I = 750
at (750, 0)
PIVOT 2
I
1 0 0 0
x
0 0 1 0
Choosing the pivot column
y
-0.3
0.5
0.5
0.5
s
1 0 1 0 0
s
2 0.5
-0.5
0.5
-1.5
s
3 0 0 0 1 RHS 750 250 750 150 Most negative number in objective row
PIVOT 2
I
1 0 0 0
x
0 0 1 0
Choosing the pivot element
y
-0.3
0.5
0.5
0.5
s
1 0 1 0 0
s
2 0.5
-0.5
0.5
-1.5
s
3 0 0 0 1 RHS 750 250 250/0.5
750 750/0.5
150 150/0.5
Ratio test: Min. of 3 ratios gives 0.5 as pivot element
PIVOT 2
I
1 0 0
0
x
0 0 1
0 Making the pivot
y
-0.3
0.5
0.5
1
s
1 0 1 0
0
s
2 0.5
-0.5
0.5
-3
s
3 0 0 0
2
RHS 750 250 750
300
Divide through the pivot row by the pivot element
PIVOT 2
I
1
0 0 0
x
0
0 1 0
y
0
0.5
0.5
1
Making the pivot
s
1
0
1 0 0
s
2
-0.4
s
3
0.6
RHS
840
-0.5
0 250 0.5
0 750 -3 2 300 Objective row + 0.3 x pivot row
PIVOT 2
I
1
0
0 0
x
0
0
1 0
y
0
0
0.5
1
Making the pivot
s
1 0
1
0 0
s
2 -0.4
1
0.5
-3
s
3 0.6
-1
RHS 840
100
0 750 2 300 First constraint row – 0.5 x pivot row
PIVOT 2
I
1 0
0
0
x
0 0
1
0
y
0 0
0
1
s
1 0 1
0
0
Making the pivot
s
2 -0.4
1
2
-3
s
3 0.6
-1 RHS 840 100
-1 600
2 300 Second constraint row – 0.5 x pivot row
PIVOT 2
I
1
0 0 0
x
0 0
1
0
y
0 0 0
1
s
1 0
1
0 0
New solution
s
2 -0.4
1 2 -3
s
3 0.6
-1 RHS
840 100
-1
600
2
300
I = 840, x = 600, y = 300, s 1 = 100, s 2 = 0, s 3 = 0
LINEAR PROGRAMMING
y
960 800
Example
Maximise I = x + 0.8y subject to x + y 1000 2x + y 1500 3x + 2y 2400 640 480 320 160 0 0 160 320 480 640 800 960
x
Solution after pivot 2:
I = 840
at (600, 300)
PIVOT 3
I
1 0 0 0
x
0 0 1 0
Choosing the pivot column
y
0 0 0 1
s
1 0 1 0 0
s
2 -0.4
1 2 -3
s
3 0.6
-1 RHS 840 100 -1 600 2 300 Most negative number in objective row
PIVOT 3
I
1 0 0 0
x
0 0 1 0
Choosing the pivot element
y
0 0 0 1
s
1 0 1 0 0
s
2 -0.4
1 2 -3
s
3 0.6
-1 RHS 840 100 100/1 -1 600 600/2 2 300 Ratio test: Min. of 2 ratios gives 1 as pivot element
PIVOT 3
I
1
0
0 0
x
0
0
1 0
y
0
0
0 1
s
1 0
1
0 0
Making the pivot
s
2 -0.4
1
2 -3
s
3 0.6
-1
RHS 840
100
-1 600 2 300 Divide through the pivot row by the pivot element
PIVOT 3
I
1
0 0 0
x
0
0 1 0
y
0
0 0 1
Making the pivot
s
1
0.4
1 0 0
s
2
0
1 2 -3
s
3
0.2
RHS
880
-1 100 -1 600 2 300 Objective row + 0.4 x pivot row
PIVOT 3
I
1 0
0
0
x
0 0
1
0
y
0 0
0
1
s
1 0.4
1
-2
0
Making the pivot
s
2 0 1
0
-3
s
3 0.2
-1 RHS 880 100
1
2
400
300 Second constraint row – 2 x pivot row
PIVOT 3
I
1 0 0
0
x
0 0 1
0
y
0 0 0
1
s
1 0.4
1 -2
3 Making the pivot
s
2 0 1 0
0
s
3 0.2
-1 RHS 880 100 1
-1
400
600
Third constraint row + 3 x pivot row
PIVOT 3
I
1
0 0 0
x
0 0
1
0
y
0 0 0
1
s
1 0.4
1 -2 3
Optimal solution
s
2 0
1
0 0
s
3 0.2
-1 RHS
880 100
1 -1
400 600
I = 880, x = 400, y = 600, s 1 = 0, s 2 = 100, s 3 = 0
LINEAR PROGRAMMING
y
960 800
Example
Maximise I = x + 0.8y subject to x + y 1000 2x + y 1500 3x + 2y 2400 640 480 320 160 0 0 160 320 480 640 800 960
x
Optimal solution after pivot 3:
I = 880
at (400, 600)